where \(a_1^+(t_{a_1},a_1)\)  and \(a_1^-(-t_{a_1}+\iota \frac{\beta }{2},a_1)\)  are the boundaries of the island surface in the right and left wedges of the black hole patch. In the large distance approximation (observer is very far away from the black hole horizon), \(d(a_1^+,a_1^-)\equiv d(b_1^+,b_1^-) \equiv d(a_1^{\pm },b_1^{\mp }) \gg d(a_1^{\pm },b_1^{\pm })\)  [1]}, REF  simplifies to
\(S_{\rm matter}^{\rm BEH}({\cal R}\cup {\cal I}) = \frac{Q}{6} \log \left(d^2(a_1^+,b_1^+)\right)\) 
